A genus six cyclic tetragonal reduction of the Benney equations

This page is a repository of results related to the following paper, published by Journal of Physics A :

M England and J Gibbons. A genus six cyclic tetragonal reduction of the Benney equations. J. Phys. A: Math. Theor. 42 (2009) 375202.

Journal website.
Preprint: arXiv:0903.5203.

Abstract: A reduction of Benney's equations is constructed corresponding to a Schwartz-Christoffel map associated with a cyclic tetragonal curve. The mapping function, a second kind Abelian integral on this Riemann surface, is constructed explicitly as a rational expression in derivatives of the Kleinian sigma function of the curve.

Authors: Matthew England (Heriot Watt University) and John Gibbons (Imperial College London).

This paper makes reference to a large number of relations, which for reasons of brevity have not been included in the paper. As such links to these relations are provided here.

The (4,5)-curve and sigma function expansion

The reduction described in the paper makes use of a cyclic tetragonal curve of genus six, which can be defined by f(x,y)=0 where

f(x,y) = y⁴ - ( x⁵ + λ₄x⁴ + λ₃x³ + λ₂x² + λ₁x + λ₀ ).

In the paper we define and work with the Kleinian sigma function that is associated with this curve. The definition and properties of this function are described in the paper. We use a power series expansion of the sigma function, that was constructed by partitioning it into polynomials whose terms have the same weight ratio. The expansion was given as

σ(u) = C₁₅ + C₁₉ + C₂₃ + ... + C_15+4n + ...

where C_k is a finite polynomial composed of products of monomials in u_i of total weight k, multiplied by monomials in λ_j of total weight 15-k.

Within the paper we use the expansion up to an including C₄₇. The polynomials are given in the text files below as functions of {v₁,v₂,v₃,v₄,v₅,v₆} and the curve constants lambda.

Click on the following file names for the text file containing that polynomial.

SW.txt. This is the Schur Weierstrass polynomial, SW₄₅, which was shown to equal C₁₅.
C19.txt
C23.txt
C27.txt
C31.txt
C35.txt
C39.txt
C43.txt
C47.txt

The series expansion of sigma(u) as it descends a strata

In Section 6 of the paper we describe how a set of relations between derivatives of the sigma-function can be constructed for each strata of the Jacobian. To derive the relations we used a series expansion of sigma(u) as u descended a strata.
To find this expansion we wrote u = hat(u) + u_xi where u is a point on Theta^[i], hat(u) is a point on Theta^[i-1] and u_xi contains the expansions of u in the local parameter xi at infinity. We then calculated the series expansion of sigma(u) in Maple. We did not use Maple's series command, but instead designed a more efficient program to take advantage of the weight structure in the function.
The expansion had been constructed to O(xi^29) as is contained in this text file.

Partial differential equations satisfied by sigma(u) on Theta^[1]

We use the above expansion to derive sets of relations between derivatives of the sigma-function. A full description of how these relations are derived is given in the paper. We use the set of relations valid for u in Theta^[1] to prove our results for the mapping function. Some of these relations were given in the Appendix, but the full sets that we derived are given below. These have been calculated using the expansion to O(xi^29).

RelThet4.txt contains those relations valid on Theta^[4].
RelThet3.txt contains those relations valid on Theta^[3].
RelThet2.txt contains those relations valid on Theta^[2].
RelThet1.txt contains those relations valid on Theta^[1].

Partial differential equations satisfied by sigma(u) at u = u_0

We denote by u_0 the point on Theta^[1] which is the preimage of the point on the curve at infinity. This is shown in the paper to coincide with the point where sigma[23](u)=0. In Section 7 we derived a further set of relations between the sigma-derivatives that were valid at this point, (but not valid on Theta^[1] in general). Some of these relations were given in the Appendix, but the full set that we derived can be found here.

The Kleinian sigma-function can also be used to define sets of Abelian functions associated with the curve. These include generalisations of the Weierstrass p-function. The following related paper deals with the theory of Abelian functions associated with the cyclic (4,5)-curve:

M England and JC Eilbeck. Abelian functions associated with a cyclic tetragonal curve of genus six. J. Phys. A: Math. Theor. 42 (2009) 095210.

A pdf of the final version of the paper is available here and this website gives links to many of the results on Abelian functions that were discussed in the above paper.

This page is maintained by Matthew England.

last updated: 24th March 2010